Mechanics is the body of knowledge that deals with the relationships between forces and the motion of points through space, including the material space. Material Science is the body of knowledge that deals with the properties of materials, including its mechanical properties. Mechanics is very deductive--having defined some variables and given some basic premises, one can logically deduce relationships between the variables. Material Science is very empirical--having defined some variables one establishes the relationships between the variables experimentally. Mechanics of Materials synthesizes the empirical relationships of materials, into the logical framework of mechanics, to produce formulas for use in the design of structures and other solid bodies.
In the past twenty-five years there has been tremendous growth in Mechanics, Material Science, and in new applications of Mechanics of Materials. Twenty-five years ago, techniques such as the Finite Element Method and Moire' Interferometry, were research topics in mechanics, but today, these techniques are routinely used in engineering design and analysis. Twenty-five years ago, wood and metal were the preferred materials in engineering design, but today, machine components and structures may be made up of plastics, ceramics, polymer composites, and metal matrix composites. Twenty-five years ago, Mechanics of Materials was primarily used for structural analysis in Aerospace, Civil, and Mechanical engineering, but today, Mechanics of Materials is used in electronic packaging, medical implants, explanation of geological movements, and the manufacturing of wood products to meet specific strength requirements. Though the principles in Mechanics of Materials have not changed in the past twenty-five years, the presentation of these principles must evolve to provide the students with a foundation that will permit them to readily incorporate the growing body of knowledge as an extension of fundamental principles and not something added on, and vaguely connected to, what they already know. This is my primary motivation for writing this book.
Often one hears arguments that seem to suggest that intuitive development comes at the cost of mathematical logic and rigor, or, the generalization of a mathematical approach comes at the expense of intuitive understanding. Yet, the icons in the field of Mechanics of Materials, such as Cauchy, Euler, and Saint-Venant, were individuals who successfully gave physical meaning to the mathematics they used. Accounting of shear stress in the bending of beams is a beautiful demonstration of how the combination of intuition and experimental observations can point the way when self-consistent logic does not. Intuitive understanding is a must--not only for creative engineering design but also for choosing the marching path of a mathematical development. By the same token, it is not the heuristic-based arguments of the older books, but the logical development of arguments and ideas that provide students with the skills and principles necessary to organize the deluge of information in modern engineering. Building a complimentary connection between intuition, observations, and mathematical generalization is central to the design of this book.
Learning the course content is not an end in itself but a part of an educational process. Some of the serendipitous development of theories in Mechanics of Materials, the mistakes made, the controversies that arose from these mistakes, are all part of the human drama that has many educational values including: learning from others mistakes, the struggle in understanding difficult concepts, and the fruits of perseverance. The connections of ideas and concepts discussed in a chapter to advanced modern techniques also has educational values including: continuity and integration of subject material, a starting reference point in a literature search, an alternative perspective, and an application of the subject material. Incorporating the above described educational values without distracting the student from the central ideas and concepts of Mechanics of Materials is an important complimentary objective of this book.
The achievement of educational objectives emphasized above is intricately tied to the degree to which the book satisfies the pedagogical needs of the students . The `Note to the Students' describes some of the features that address the pedagogical needs of the students. The `Note to the Instructor' describes the design and format of the book to meet the above described objectives.
I welcome any comments, suggestions, concerns, or corrections you may have that will help me improve the book. You could either relay your input to the publisher or to me. My e-mail address is: mavable@mtu.edu.
Described below are some of the features that should help you meet the learning objectives of the book.
A course in Statics is pre-requisite for this course. Appendix A reviews the concepts of Statics briefly from the perspective of this course. If you had Statics few terms ago, then you may need to review your Statics textbook before the brevity of presentation in Appendix A serves you adequately. If you feel comfortable with your knowledge of Statics then you can assess for yourself what you need to review by using the Statics Review Exams that are given in Appendix A.
All internal forces and moments are in bold italics . This is to emphasize that the internal forces / moments must be determined by making an imaginary cut, drawing free body diagram, and using equilibrium equations or by using methods that are derived from this approach.
Every chapter starts with the section of Overview which describes the motivation for studying the chapter and the two major learning objectives in the chapter.
Every chapter has a section, Closure, that serves as a connecting link to topics in subsequent chapters. Of particular importance are `Closure' in chapters three and seven as these are the two links connecting the three major parts of the book together.
Every chapter ends with Points and Formulas to Remember, a one page synopsis of non-optional topics. This brings greater focus to the material that must be learned.
Every Example problem starts with a Plan and ends with Comments, both of which are indented to emphasize the importance of these two features. Developing a Plan before solving a problem is essential for the development of analysis skills. Comments are, observations deduced from the example highlighting concepts discussed in the text before the example, or, observations that suggests the direction of development of concepts in the text after the example.
Quick-tests with solutions are designed to help you diagnose your understanding of the text material. To get the maximum benefit of these tests, take these tests only after you feel comfortable with your understanding of the text material.
After a major topic you will see a box called Consolidate your Knowledge that will suggest to you to either write a synopsis or derive a formula. Consolidate your Knowledge is a learning device that is based on the observation that it is easy to follow someone else's reasoning but significantly more difficult to develop one's own reasoning. By deriving a formula with the book closed or by writing a synopsis of the text, you force yourself to think of details you would not otherwise. When you know your material well, writing will be easy and will not take much time.
Every chapter has a section called General Information where connections of the chapter material to historical development and advanced topics is made. History shows that concepts are not an outcome of linear logical thinking but rather a struggle in the dark in which mistakes were often made but the perseverance of pioneers has left us with a rich inheritance. Connection to advanced topics is an extrapolation of the concepts studied. Other reference material that may be helpful in the future can be found in problems labeled `Stretch Yourself'.
I hope you will enjoy reading this book as much as I did writing it.
The best way by which I can show you how the presentation in this book meets the objectives stated in the `Preface' is by drawing your attention to certain specific features. Described below is the underlying design and motivation of presentation in context of development of theories of one-dimensional structural elements and the concept of stress. But the same design philosophy and motivation permeates the rest of the book.
depicts the logic relating displacements--strains--stresses--internal-forces / moments--external-forces / moments. The logic is intrinsically very modular-- equations relating the fundamental variables are independent of each other, hence complexity can be added at any point without affecting the other equations. This is brought to the attention of the reader in , where the stated problem is to determine the force exerted on the car carrier by a stretch cord holding a canoe in place. The problem is first solved as a straight forward application of the logic shown in . Then, in comments following the example, it is shown how different complexities (in this case non-linearities) can be added to improve the accuracy of the analysis. Associated with each complexity is a post-text problem (number written in parenthesis) that is under the heading `Stretch Yourself' or `Computer Problems' that are well within the scope of the student willing to stretch themselves. Thus, the central focus in is learning the logic of which is fundamental to Mechanics of Materials. But the student can appreciate how complexities can be added to simplified analysis, even if no `Stretch Yourself' problems are solved.
The above philosophy, used in , is also used in developing the simplified theories of axial members, torsion of shafts, and bending of beams. The development of the theory for structural elements is done rigorously, with assumptions identified at each step. A footnote associated with an assumption directs the reader to examples, optional sections, and `Stretch Yourself' problems, where the specific assumption is violated. Thus, in on theory of torsion of shafts: of linear-elastic material has a footnote directing the reader to see `Stretch Yourself' problems and for non-linear material behavior; of Material Homogeneity across the cross-section, has a footnote directing the reader to see the optional on Composite Shafts; of untapered shaft is followed by statements directing the reader to on tapered shaft. shows the synopsis of all three (axial, torsion and bending) theories on a single page, to show the underlying pattern in all theories in Mechanics of Materials that the student have seen three times. The central focus in all three cases remains on the simplified basic theory, but the presentation in this book should help the students develop an appreciation of how different complexities can be added to the theory, even if no `Stretch Yourself' problems are solved or optional topics covered in class.
Compact organization of information seems like an abstract reason for learning theory to some engineering students. Some students have difficulty visualizing a continuum as an assembly of infinitesimal elements whose behavior can be approximated and / or deduced. There are two features in the book that address these difficulties. I have included a section called Prelude to Theory in `Torsion of Circular Shafts' and `Symmetric Bending of Beams'1. In `Prelude to Theory', numerical problems are considered in which discrete bars welded to rigid plates are considered. The rigid plates are subjected to displacements that simulate the kinematic behavior of cross-sections in torsion or bending. Using the logic of , the problems are solved--effectively developing the theory in a very intuitive manner, and the section on `Theory' is essentially formalizing the observations of numerical problems in `Prelude to Theory'. The second feature are actual photographs showing un-deformed and deformed grids due to axial, torsion and bending loads. `Seeing is believing' is better than accepting on faith that a drawn deformed geometry represents the actual situation. In this manner the complimentary connection between intuition, observations, and mathematical generalization is achieved in the context of one-dimensional structural elements.
Double subscripts2 are used with all stresses and strains. The use of double subscripts has three distinct benefits: (i) It provides students with a procedural way to compute the direction of a stress component which they calculate from a stress formula. The procedure of using subscripts is explained in and elaborated in . This procedural determination of direction of a stress component on a surface can help many students overcome the shortcomings in intuitive ability. (ii) Computer programs such as Finite Element Method or those that reduce full-field experimental data, produce stress and strain values in a specific coordinate system that must be properly interpreted, which is possible if the students know how to use subscripts in determining the direction of stress on a surface. (iii) It is consistent with what the student will see in more advanced courses such as those on composites where the material behavior can challenge many intuitive expectations.
But it must be emphasized that the use of subscripts is to compliment not substitute , intuitive determination of stress direction. Procedures for determining the direction of a stress component by inspection and by subscripts are briefly described at end of each theory section of structural elements and examples such as: on axial members, and on torsional shear stress, on bending normal stress emphasize both approaches. Similarly there are sets of problems in which stress direction must be determined by inspection as there are no numbers given --problems such as: through on direction of torsional shear stress; through on tensile and compressive nature of bending normal stress; through on the direction of normal and shear stress on an inclined plane. If subscripts are to be successfully used in determining the direction of a stress component obtained from a formula, then the sign conventions for drawing internal forces / moments on free body diagram must be followed, hence there are examples (such as ) and problems (such as and ) in which the sign of internal quantities are to be determined by sign conventions. Thus, once more, the complimentary connection between intuition and mathematical generalization is enhanced by using double subscripts for stresses and strains.
Other features that you may find useful are described briefly below.
All optional topics are marked by an asterisk (*) to account for instructor interest and pace. Skipping these topics can at most affect the student ability to solve some post-text problems in subsequent chapters and these problems are easily identifiable.
`General Information' in all chapters is an optional section. In some examples and post-text problems, reference is made to a topic that is described in `General Information'. The only purpose of this reference is to draw attention to the topic but knowledge about the topics is not needed for solving the problem.
The topics of stress and strain transformation can be moved before the discussion of structural elements (Chapter 4) as shown in the two syllabi in the instructor's manual. I have strived to eliminate confusion regrading maximum normal and shear stress at a point with the maximum values of stress components calculated from the formulas developed for structural elements.
The post-text problems are categorized for ease of selection for discussion and / or assignments. Generally speaking, the starting problems in each problem set are single concept problems. This is particularly true in the later chapters where problems are designed to be solved by inspection to encourage development of intuitive ability. ` Design type ' problems, involving sizing of members, selection of materials (later chapters) to minimize weight, determination of maximum allowable load to fulfil one or more limitations on stress or deformation, and construction and use of failure envelopes in optimum design (Chapter 10). ` Stretch Yourself ' problems are optional problems for motivating and challenging students who spend time and effort understanding the theory. These problems often involve extension of the theory to include added complexities. ` Computer problems ' are also optional problems and require knowledge of spread sheets, or, simple numerical methods such as: numerical integration, roots of a non-linear equation in some design variable, or use of Least square method. Additional categories such as: ` Stress Concentration Factor Problems ', ` Stress Intensity Factor Problems ', ` Fatigue Problems ', and ` Transmission of Power Problems ' are chapter specific optional problems associated with optional text sections.
The book can be viewed as made up of three major parts. The first part consists of the first three chapters, where in addition to introducing the basic concepts of stress, strain, and Hooke's law, an important underlying theme is to built the links of the logic of relating displacements to external forces that is shown in . The second part consists of chapters four through seven, where the theories and formulas of one-dimensional structural elements are developed and used in analysis and design of these structural elements. The third part consists of chapters eight through ten in which stress and strain transformation are prelude to chapter ten on design and failure of structures made from one-dimensional elements. The section `Closure' in chapter four and seven are important connecting links between the three parts.
I have tried to make a mix of practical problems as shown by photographs, and problems that are designed for understanding the fundamental principals. If you can suggest other variety of problems that will be useful in teaching this course material, then please let me know. I will gratefully acknowledge your contribution.
1. I did develop a similar section for axial members, but found most of the ideas had been covered in . So in Axial members the very first example is on the logic of , and the first four post-text problems are also based on it. An instructor could chose to cover this material before developing the theory for axial members as is done in torsion and bending.